Bit nozzle selection - Optimized hydraulics for two and three jets

Input(s)

\(N_{a}\): Jet Size for nozzle 1 (in.)

\(N_{b}\): Jet Size for nozzle 2 (in.)

\(N_{c}\): Jet Size for nozzle 3 (in.)

Ca: Circulation Rate (gpm)

\(\mathrm{Cb}\): Circulation Rate (gpm)

Wm: Mud Weight (ppg)

Cpa: Circulating Pressure 1 (psi)

\(\mathrm{Cpb}\): Circulating Pressure \(2(\mathrm{psi})\)

\(P_{\max }\): Max surface pressure (psi)

\(\mathrm{d}\): Dia of the nozzle chosen (in.)

\(\mathrm{n}\): No. of nozzles (unit)

Output(s)

A: Nozzle Area (in. \({ }^{2}\) )

Pba: Bit Nozzle Pressure Loss 1 (psi)

Pbb: Bit Nozzle Pressure Loss 2 (psi)

\(P_{c a}\): Total Pressure Loss for pump pressure 1 except bit nozzle (psi)

\(P_{c b}\): Total Pressure Loss for pump pressure 2 except bit nozzle (psi)

M: Slope of Line (unitless)

\(P_{\text {iopt }}\): Optimum Pressure for Impact Force (psi)

\(P_{\text {hopt }}\): Optimum Pressure for hydraulic Horsepower (psi)

Pilb: Pressure loss at the bit due to impact force (psi)

Phlb: Pressure loss at the bit due to hydraulic horsepower (psi)

\(Q_{\text {iopt }}\): Optimum flowrate for impact force (gpm)

\(Q_{\text {hopt }}\): Optimum flowrate for hydraulic horsepower (gpm)

\(A_{\text {in }}\): Area of nozzle for Impact force (in. \({ }^{2}\) )

\(A_{h n}\): Area of nozzle used for hydraulic horsepower (in. \({ }^{2}\) )

\(A_{n}\): Area of nozzle used (in. \({ }^{2}\) )

Formula(s)

\[ \begin{aligned} & \mathrm{A}=\frac{\mathrm{N}_{\mathrm{a}}^{2}+\mathrm{N}_{\mathrm{b}}^{2}+\mathrm{N}_{\mathrm{c}}^{2}}{1303.8} \\ & P b a=(C a)^{2} * \frac{W m}{10858 * A} \\ & P b b=(C b)^{2} * \frac{W m}{10858 * A} \\ & P_{c a}=C p a-P b a \\ & P_{c b}=C p b-P b b \\ & M=\frac{\log \left(\frac{P_{c a}}{P_{c b}}\right)}{\log \left(\frac{C a}{C b}\right)} \\ & P_{\text {iopt }}=2 * \frac{P_{\max }}{M+1} \\ & P_{\text {hopt }}=2 * \frac{P_{\max }}{P_{\max }+1} \\ & \text { Pilb }=\left(\frac{P_{\text {iopt }}}{P_{\text {max }}}\right)^{1+M} * C a \\ & \text { Phlb }=\left(\frac{P_{\text {hopt }}}{P_{\text {max }}}\right)^{1+M} * C a \\ & Q_{\text {iopt }}=P_{\text {max }}-P_{\text {iopt }} \\ & Q_{\text {hopt }}=P_{\text {max }}-P_{\text {hopt }} \\ & A_{\text {in }}=\left(Q_{\text {iopt }}^{2} * \frac{W m}{10858 * P_{\text {max }}}\right)^{0.5} \\ & A_{\text {hn }}=\left(Q_{\text {hopt }}^{2} * \frac{W m}{10858 * P_{\text {max }}}\right)^{0.5} \\ & A_{n}=d *\left(\frac{A}{n * 0.7854}\right)^{0.5} \end{aligned} \]

Reference(s)

Formulas and Calculations for Drilling, Production and Workover, Second Edition, Lapeyrouse, Page: 166.

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